Our results reveal the importance of directed interactions for the long-term prospects of cooperative behavior in a population.
Our analysis also highlights an interesting relation between the direction of social interactions versus the direction of behavioral spread (or dispersal). We have focused on two opposite extremes: downstream dispersal, in which behavior spreads in the same direction as directed social interactions, and upstream dispersal, in which behavior spreads in the opposite direction. In the downstream setting, the directional motif \(i\rightarrow j\)  and \(j\rightarrow \ell \)  indicates that a cooperator \(i\)  donates to \(j\) , and \(j\)  might then imitate \(i\) 's cooperative behavior and make a donation to \(\ell \) . This form of dispersal is roughly analogous to generalized reciprocity — a player who received help from another player feels motivated to help the third player in turn [1]}. In the upstream setting, by contrast, the directional motif \(i\rightarrow j\)  and \(\ell \rightarrow i\)  indicates that cooperator \(i\)  donates to \(j\) , and \(\ell \)  might then imitates \(i\) 's cooperative behavior and make a donation to \(i\) . This form of behavioral dispersal is roughly analogous to indirect reciprocity — the player who helped another player receives a benefit from a third party [2]}.
Despite these rough similarities and their beneficial effects for cooperation, the form of reciprocity arising from directed dispersal of behavior differs from generalized reciprocity and from indirect reciprocity: the dispersal mechanism is payoff-dependent while the later mechanisms are action-dependent. A synthetic understanding of reciprocity based on conditional behavior versus reciprocity achieved by payoff-biased imitation remains an important topic for future research.
